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Publication# Evolution of open many-electron systems

Abstract

Starting from the quantum statistical master equation derived in [1] we show how the connection to the semi-classical Boltzmann equation (SCBE) can be established and how irreversibility is related to the problem of separability of quantum mechanics. Our principle goal is to find a sound theoretical basis for the description of the evolution of an electron gas in the intermediate regime between pure classical behavior and pure quantum behavior. We investigate the evolution of one-particle properties in a weakly interacting N-electron system confined to a finite spatial region in a near-equilibrium situation that is weakly coupled to a statistical environment. The equations for the reduced n-particle density matrices, with n < N are hierarchically coupled through two-particle interactions. In order to elucidate the role of this type of coupling and of the inter-particle correlations generated by the interaction, we examine first the particular situation where energy transfer between the N-electron system and the statistical environment is negligible, but where the system has a finite memory. We then formulate the general master equation that describes the evolution of the coarse grained one-particle density matrix of an interacting confined electron gas including energy transfer with one or more bath subsystems, which is called the quantum Boltzmann equation (QBE). The connection with phase space is established by expressing the one-particle states in terms of the overcomplete basis of coherent states, which are localized in phase space. In this way we obtain the QBE in phase space. After performing an additional coarse-graining procedure in phase space, and assuming that the interaction of the electron gas and the bath subsystems is local in real space, we obtain the semi-classical Boltzmann equation. The validity range of the classical description, which introduces local dynamics in phase space is discussed.

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Semiconductor quantum dots are usually compared to artificial atoms, because their electronic structure consists of discrete energy levels as for natural atoms. These artificial systems are integrated in solid materials and can be localized with a spatial precision of the order of nanometers. Besides, they conserve their quantum properties even at quite high temperatures (∼ 10 K). These properties make quantum dots one of the most suitable systems for the realization of quantum devices and computers. However, the energy states and the optical properties of quantum dots are much more complicated than for atoms, because a quantum dot is never an isolated potential well. Instead, its electronic structure depends on the crystallin structure of the semiconductor material and on the Coulomb ion-electron and electron-electron correlations. In particular, the presence of several valence bands and their mixing, induced by quantum confinement, gives rise to novel properties which are still not completely understood and exploited in applications. To get a major advance in this field, a full deterministic control of the spatial shape of the quantum confinement is needed, combined with a deeper understanding of the connections between electronic and optical properties. This thesis work has these two main objectives. We realized and experimentally studied different quantum dot systems, in pyramidal hetero-structures grown with MOCVD techniques. These systems allowed the realization of several different geometries for the carrier confining potential, with a precision in the order of nanometers. The optical characterization has been obtained in particular by means of polarization-resolved microphotoluminescence, magneto-photoluminescence, excitation photoluminescence (PLE), and interferometry techniques. For single quantum dots, we have observed and characterized for the first time new excitonic complexes, arising from excited hole states. This allowed a full caracterizatioon of the valence band hole states in our peculiar system. By means of photon correlation measurements, we have also experimentally demonstrated that, even in presence of a large family of exciton states, these quantum dot systems can emit single photons. We have then realized much more complex quantum dot structures, double dot systems (quantum dot molecules) and a completely new system called Dot-in-Dot (DiD). This latter is composed by a small inner dot surrounded by an electrostatic potential well (which can be considered as an outer elongated dot). Such a composite system is characterized by a strong valence band mixing. This state superposition is however very sensitive to small variations of the confining potential. Therefore the degree of valence band mixing can be easily switched by the introduction of a week external field. Since the valence band mixing determines the polarization properties of the emitted light, the DiD changes the polarization properties of its emission spectrum under the action of an external field. In particular, we have experimentally demonstrated this effect for an external static magnetic field, while we have numerically predicted a very similar effect for a static electric field. In the latter case, the polarization switching is a direct consequence of the quantum confined Stark effect induced in the DiD. Hence the DiD appears to be an ideal candidate for realizing emitters of single photons with tunable and controllable polarization.

The exploration of open quantum many-body systems -systems of microscopic size exhibiting quantum coherence and interacting with their surrounding- has emerged as a key research area over the last years. The recent advances in controlling and preserving quantum coherence at the level of a single particle, developed in a wide variety of physical platforms, have been a major driving force in this field. The driven dissipative nature is a common characteristic of a wide class of modern experimental platforms in quantum science and technology, such as photonic systems, ultracold atoms, optomechanical systems, or superconducting circuits. The interplay between the coherent quantum dynamics and dissipation in open quantum systems leads to a wide range of novel out-of-equilibrium behaviours. Among them, the emergence in these systems of dynamical phases with novel broken symmetries, topological phases and the occurrence of dissipative phase transitions are of particular interest. This thesis aims at establishing a theoretical framework to engineer, characterize and control nonclassical states of light in photonic quantum optical networks in different regimes. The emphasis is put on its implementation, in particular with respect to integration and scalability in photonic platforms. In this thesis, we tackle some interesting aspects arising in the study of the dynamics of driven dissipative coupled nonlinear optical resonators. In that context, we consider the dynamics of two coupled nonlinear photonic cavities in the presence of inhomogeneous coherent driving and local dissipations using the Lindblad master equation formalism.We show that this simple open quantum many-body system can be subject to dynamical instabilities. In particular, our analysis shows that this system presents highly nonclassical properties and its dynamics exhibits dissipative Kerr solitons (DKSs), characterized by the robustness of its specific temporal or spatial waveform during propagation.In a second step, our intuition gained from this system composed of only few degrees of freedom is expanded to the study of systems of bigger size. In particular, we study DKSs originating from the parametric gain in Kerr microresonators. While DKSs are usually described using a classical mean-field approach, our work proposes a quantum-mechanical model formulated in terms of the truncated Wigner formalism. This analysis is motivated by the fact that technological implementations push towards the realization of DKSs in miniaturized integrated systems. These are operating at low power, a regime where quantum effects are expected to be relevant. Using the tools provided by the theory of open quantum systems, we propose a detailed investigation of the impact of quantum fluctuations on the spectral and dynamical properties of DKSs. We show that the quantum fluctuations arising from losses engender a finite lifetime to the soliton, and demonstrate that DKSs correspond to a specific class of dissipative time crystals.

Jean-Philippe Ansermet, François Reuse, Jacques Van Der Klink

We present a detailed discussion of the evolution of a statistical ensemble of quantum mechanical systems coupled weakly to a bath. The Hilbert space of the full system is given by the tensor product between the Hilbert spaces associated with the bath and the bathed system. The statistical states of the ensemble are described in terms of density matrices. Supposing the bath to be held at some - not necessarily thermal - statistical equilibrium and tracing over the bath degrees of freedom, we obtain reduced density matrices defining the statistical states of the bathed system. The master equations describing the evolution of these reduced density matrices are derived under the most general conditions. On time scales that are large with respect to the bath correlation time τB corr and with respect to the reciprocal transition frequencies of the bathed system, the resulting evolution of the reduced density matrix of the bathed system is of Markovian type. The detailed balance relations valid for a thermal equilibrium of the bath are derived and the conditions for the validity of the fluctuation-dissipation theorem are given. Based on the general approach, we investigate the non-linear response of the bathed subsystem to a time-periodic perturbation. Summing the perturbation series we obtain the coherences and the populations for arbitrary strengths of the perturbation. © 2003 Springer-Verlag Berlin/Heidelberg.

2003